\(D(\mathbf {x},\sigma )\)  provides a close approximation to \(\sigma ^2\Delta L(\mathbf {x},\sigma )\) , the normalized Laplacian of \(L\) .
We refer to [1]} for more details and a rigorous analysis including the stability of the keypoint computation. The seminal paper introducing SIFT [2]} sparked an explosion of local keypoints detectors and descriptors seeking discrimination and invariance to specific groups of image transformations [3]}.
Mikolajczyk [4]} found that the maxima and minima of the operator \(\sigma ^2\Delta L(\mathbf {x},\sigma )\)  produce the most stable image keypoints compared to a range of other possible image operators, such as the gradients, Hessian, or Harris corner detector. The SIFT keypoints have been used in countless image processing and computer vision applications, such as image registration [5]}, [6]}, camera calibration [7]}, 3D reconstruction [8]}, object recognition [9]}, [10]}, [11]}, [12]}.
